text stringlengths 16 3.88k | source stringlengths 60 201 |
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18.725 Algebraic Geometry I
Lecture
4
Lecture 4: Grassmannians, Finite and Affine Morphisms
Remarks on last time
1. Last time, we proved the Noether normalization lemma: If A is a finitely generated k-algebra, then, A
contains B ∼= k[x1, . . . , xn] (free subring) such that A is a finitely generated B-module.
Question: Whe... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
a = , b = , (a = b2)
X
X
Y
∼= A1 = U1
Note that U1 ∩ U2
∼= A1 \ {0}.
By changing coordinates, we can take the degree 2 curve in P2 to be X 2 + Y 2 = Z 2. Connect points in
a quadric to a fixed point. In practice, we can work with the point (1 : 0 : 1). We identify the set of all
lines through a given point with P1. Taki... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
X =
n
(cid:91)
i=1
Xi, where the Xi are closed irreducible subsets of X. Without loss of generality, we
can assume that none of the Xi are a subset of another. Then, Xi is not a subset of
(cid:91)
Xj (follows from
j
=i
j ∩ Xi. Since every
irreducibility). Otherwise, we would have that Xi is a union of proper closed sub... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
ible. Thus, we must either
have f = 0 or g = 0 and A has no zerodivisors.
Conversely, suppose that Spec A is not irreducible. Let X = Spec A. Then, we can write X = Z1 ∪ Z2,
where Z1, Z2 (cid:40) X are proper closed subsets of X. Since proper closed subsets correspond to nonzero ideals,
we can pick nonzero f ∈ IZ1 and ... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
��nes a projective algebraic variety. The embedding of Gr(k, n) into projective space
is defined by W (cid:55)→ the line W ⊂ V .
k
(cid:94)
k
(cid:94)
Claim: This map realizes Gr(k, n) as a closed subvariety in P
(cid:33)
(cid:32)
k
(cid:94)
V
(n)
= P k −1.
Example 2. Consider the case n = 4 and k = 2. These are lines i... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
or p. 42 – 44 (in 3rd edition) in Section 1.4.1 (“Closed Subsets of Projective Space”)
of Basic Algebraic Geometry 1 by Igor Shafarevich.
Finite and affine morphisms
Definition
(cid:91)
Y = Ui where the Ui are affine open pieces such that the f −1(Ui) ⊂ X are affine.
1. A morphism of algebraic varieties f : X −→ Y is called a... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
proof of the lemma (use similar ideas as last time) (compare with Lemma 2.4.3 on p.
19 of Kempf).
Proof. Let f : X −→ Y be a finite map. We can assume X and Y are affine (statement local on line).
Since the composition of two finite maps is finite, we can also assume that Z = X. Write X = Spec A and
Y = Spec B and let I = A... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
tf ) −→ Spec A.
Example 4. The morphism A2 \ {0} −→ A2 is not affine. This is similar to an exercise in the homework
(Problem 3 of Problem Set 1). It actually follows from this and the exactness of localization. Let U ⊂ A2
be an open neighborhood of 0 such that U = A2 \ Zf for some f . Since k[U ] = k[U \ {0}], U \ {0} i... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
is only for a given
chain.
Here are some facts about the dimension of a Noetherian topological space:
• dim An = n
• If X =
n
(cid:91)
i=1
Ui, then dim X = max dim Ui.
i
• If f : X −→ Y is a finite and surjective morphism, then dim X = dim Y .
4
(cid:54)
(cid:54)
(cid:54)
MIT OpenCourseWare
http://ocw.mit.edu
18.725 Al... | https://ocw.mit.edu/courses/18-725-algebraic-geometry-fall-2015/2b05c3fbc5140429fa1e2ec6b95b355a_MIT18_725F15_lec04.pdf |
Lecture 4
Removable Singularity Theorem
Theorem 1 Let u be harmonic in Ω \ {x0}, if
�
u(x) =
|
2−n)
o( x − x0
|
|
|
o(ln x − x0 )
, n > 2,
, n = 2
as x
→
x0, then u extends to a harmonic function in Ω.
Proof: Without loss of generality, we can assume Ω = B(0, 2), then u|∂B(0,1) is contin
uous. Thus by Poisson ... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2b09fee8b8ddeca30c93dbdf5682c679_lec4.pdf |
(x) ≤ u(x), ∀x ∈ B1(0) \ {0},
thus v(x) = u(x), ∀x ∈ B1(0) \ {0}.
Now we can define u(0) = v(0), and extend u to be a harmonic function on B(0, 1),
thus a harmonic function on Ω = B(0, 2).
�
Example This gives an example of Dirichlet problem that is NOT solvable:
Take Ω = B(0, 1) \ {0}, then ∂Ω = ∂B(0, 1) ∪ {0}. C... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2b09fee8b8ddeca30c93dbdf5682c679_lec4.pdf |
degree k restricted
to Sn−1, then ΔSn−1 B(θ) = −k(k + n − 2)B(θ).
Remark 1 Let Pk be the set of homogeneous polynomials of degree k on Rn , Hk be the
set of harmonic homogeneous polynomials of degree k on Rn, then
It’s not hard to prove
P
k =
H
2
k ⊕ r
P
k−2.
dimPk =
(k + n − 1)!
,
k!(n − 1)!
so
dimHk =
(k + ... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2b09fee8b8ddeca30c93dbdf5682c679_lec4.pdf |
− n + 2, if k = 0, then p = 2 − n and B(θ) = constant, thus
u = c · r2−n, which is the fundamental solution. if k > 0, then p < 2 − n, note that
B(θ) is defined on the compact set Sn−1, thus B is bounded, so u grows faster than
the fundamental solution near the origin.
From above we get a degree gap of harmonic func... | https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/2b09fee8b8ddeca30c93dbdf5682c679_lec4.pdf |
Engineering Systems
Doctoral Seminar
ESD.83 – Fall 2011
Class 1
Faculty: Chris Magee and Joe Sussman
TA: Rebecca Kaarina Saari
Guest: Professor Joel Moses, Institute Professor,
(EECS and ESD)
© 2010 Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
1 Session 1: Overvie... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/2b0a59c98ad886ad3cebf9028e039e4b_MITESD_83F11_lec01a.pdf |
acy: Understanding of core concepts and principles – base
level of literacy on the various aspects of engineering systems
Inter-disciplinary capability: The capability to reach out to adjacent
fields in a respectful and knowledgeable way and the ability to engage
with other ES scholars in assessing the importance ... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/2b0a59c98ad886ad3cebf9028e039e4b_MITESD_83F11_lec01a.pdf |
-The
Incorporation of Social Science and Engineering into
Research and Problem Solving
Guest: Joel Moses
Class 2: How do we know what we know? - The
Generation of Cumulative Knowledge in a Field
Guest: David Kaiser
Class 3: Modeling Paradigms: Useful models and
various modeling approaches
Guest: Oli de Weck
C... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/2b0a59c98ad886ad3cebf9028e039e4b_MITESD_83F11_lec01a.pdf |
Technology
9 Assignment Summary (syllabus)
1. Observations, Data Sources and Data Reduction
Assignment (no more than 1000 word paper, 10% of the
total)
Students will be expected to select and read an NBER working
paper from a faculty-provided list and to prepare a no more
than 1000 word paper, performing a critic... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/2b0a59c98ad886ad3cebf9028e039e4b_MITESD_83F11_lec01a.pdf |
10% of the total) [assigned Session 3, due
due on Session 10]
© 2009 Chris Magee, Engineering Systems Division, Massachusetts Institute of Technology
12 Assignment Summary (syllabus) 4
8. Developing a Well-Posed Research Question*** (750-
word paper, 10% of the total)[Session 8 - due Session 12]
9. In-Depth Paper ... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/2b0a59c98ad886ad3cebf9028e039e4b_MITESD_83F11_lec01a.pdf |
to
bridge- specialization effects & ES
Are there significant differences
among different sciences?
© 2009Chris Magee and Joseph Sussman, Engineering Systems Division, Massachusetts Institute of Technology
15 Relationships among fields of
knowledge 2
There are significant differences in different fields
with “h... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/2b0a59c98ad886ad3cebf9028e039e4b_MITESD_83F11_lec01a.pdf |
-based theory across
disciplines..”
Wilson argues that each discipline should be
“consilient” with established science in other
disciplines.
He distinguishes between examples comparing
consilience by reduction (dissect a phenomenon into
its components) and consilience by synthesis
(predicting higher-order phe... | https://ocw.mit.edu/courses/ids-900-doctoral-seminar-in-engineering-systems-fall-2011/2b0a59c98ad886ad3cebf9028e039e4b_MITESD_83F11_lec01a.pdf |
Notes -on double integrals.
(Read 11.1-11.5 of Apostol.)
Just as for the case of a single integral, we have the
following condition for the existence of a double integral:
Theorem 1 (Riemann condition). Suppose f -is defined -on
Q = [arb] x [c,d]. Then f -is integrable -on Q - -if and only -if
given any E > 0, th... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
Q, - -
and i f
f - and g - a r e i n t e g r a b l e
TO prove t h i s theorem, one f i r s t v e r i f i e s t h e s e r e s u l t s
f o r s t e p f u n c t i o n s (see 1 1 . 3 ) , and t h e n u s e s t h e Riemann condi-
t i o n t o prove them f o r g e n e r a l i n t e g r a b l e f u n c t i o n s . The p r o... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
to this partition. Furthermore,
one adds the earlier inequalities to obtain
Finally, we compute
this computation uses the fact that linearity has already been
proved for step functions. ~ h u s JJQ (f + g) exists. TO
calculate this integral, we note that
by definition. ' Then
here again we use the linearity of th... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
the double integral ,b f *
Choose step functions s (x,y) and t(x,y) , defined on
Q, such that s(x,y) C f (x,y) t(x,y), and
his w e can do because
f exists. For convenience, choose
s and t so they are constant on the ?artition lines. (This
does not affect their double integrals.) Then the one-dimen-
sional integr... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
, for all y in [c,d] ,
.-
Thus S and T are step functions lying beneath and above A,
respectively, Furthermore
(see p. 3561, so that
..
j
)
It f o l l o w s t h a t
A ( y ) d y exists, by t h e Riemann c o n d i t i o n .
Now t h a t w e know A(y)
i s i n t e g r a b l e , w e can conclude
from a n e a r l ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
b a s i c q u e s t i o n s , t h e
one concerning t h e e x i s t e n c e of t h e double i n t e g r a l . W e r e a d i l y
prove t h e f o l l o w i n g :
i
Theorem 4 . - The i n t e g r a l
on t h e r e c t a n g l e Q.
continuous - -
f
e x i s t s - i f f - i s
~ r o o f . A l l one needs i s t h e smal... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
.
( 3 ) A v e r t i c a l l i n e segment h a s c o n t e n t zero.
( 4 ) A s u b s e t o f a set o f c o n t e n t z e r o h a s c o n t e n t zero.
(5) A f i n i t e union o f sets o f c o n t e n t z e r o h a s c o n t e n t zero.
(6) The graph o f a c o n t i n u o u s f u n c t i o n
y = $(x);
a < x <.b
i ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
less than E ' . Consider t h e
r e c t a n g l e s
f o r i = l,...,n.
( 1 - Q x
i 1 < E whenever x
1
The t o t a l a r e a of t h e r e c t a n g l e s Ai
They cover t h e graph of
Q,
because
i s i n the i n t e r v a l
,xi] .
( x ~ - x ) 2 € ' = 2 c 1 ( b
i-1
i=l
e q u a l s
- a ) .
T h i s number e q ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
a i n
Note t h a t t h i s lemma does n o t s t a t e merely t h a t D i s
c o n t a i n e d -i n t h e union o f f i n i t e l y many s u b r e c t a n g l e s of t h e par-
t i t i o n having t o t a l a r e a l e s s t h a n E,
b u t t h a t t h e sum of t h e
a r e a s o f -a l l t h e s u b r e c t a n g l e ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
t i t i o n .
Note t h a t by c o n s t r u c t i o n , t h e r e c t a n g l e Aj;
i s p a r t i , t i o n e d
by P ,
s o t h a t it i s a union of s u b r e c t a n g l e s Q i j
of P.
Now i f a s u b r e c t a n g l e Qij
t h e n
I n t A1; f o r some k t s o that it a c t u a l l y
c o n t a i n s a p o i n... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
- If f - is bounded - on Q, and is continuous
-on Q except on - - - -a set of content - -zero, then I/', f exists.
Proof. S t e ~ 1.We prove a preliminary result:
Suppose that given e > 0, there exist functions g and h that are integrable over Q, such
that
and
g(x) I f(x) l h(x)
for x in Q
Then f is integrable ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
of D, set
g(x) = f(x) = h(x)
for x E Q. .. Do this for each such subrectangle. Then for any other x in Q, set
1J
g(x) = -M
and h(x) = M.
T h e n g S f S h o n Q .
Now g is integrable over each subrectangle Q.. that does not contain a point of D,
1J
since it equals the continuous function f there. And g is inte... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
of D, set g(x) = f(x) = 0 and h(x) = f(x) = 0 on Q. .. Do this for each
1J
such subrectangle. For any other x in Q, set
T h e n g S f s h o n Q .
g(x) = -M
and h(x) = M.
I
Now g and h are step functions on Q, because they are constant on the interior of
each subrectangle Q. .. We compute
1J
JJQ
h = M (1(area... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
t h e s p e c i a l
f d e f i n e d on a bounded
f o r a f u n c t i o n
c a s e where S
i s a r e g i o n of Types I o r 11. We d i s c u s s h e r e
t h e g e n e r a l c a s e .
F i r s t , w e prove t h e f o l l o w i n g b a s i c e x i s t e n c e theorem:
Theorem 9. - L e t S be - - a bounded s e t i n t... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
s o continuous a t each p o i n t xl
of t h e e x t e r i o r of S, because it equals z e r o on an open b a l l
about xl. The only p o i n t s where
can' fail t o b e continuous
are p o i n t s of ths boundary of S , and this set, by assumption,
has c o n t e n t zero. Hence f!
e x i s t s . El
Q
-Note: Adjoi... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
both integrals e x i s t .
i
(c) Additivity. Let S = SL U S2. If S1 n S2 has content
zero, then
provided the right side exists.
Proof. (a) Given f, g defined on S, let 2, g equal
f t gr respectively, on S and equal 0 otherwise. Then
cl + dg equals cf + dg on S and 0 otherwise. Let Q
be a rectangle containing S... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
regions Sl and S2 that
i
intersect in a set of content zero. Since S1 is of type I
and S2 is of type 11, we can compute the integrals I[
and {I
s,
f by iterated integration. We add the results to
s,A.
f
obtain
-Area.
We can now construct a rigorous theory of area. We
already have defined the area of the rect... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
Bd S) .
Proof. Let Q be a rectangle containing S and T.
Let
is(x) = 1 for x E S
= 0 for x ft S.
Define FT similarly.
(1) If S is contained in T, then $ (x) C lT(x) .
Then by the comparison theorem,
area s = Ifs 1 = 11, L < /I,%
= j'b I. = area T.
(2) Since 0 < 1, we have by the comparison theorem,
0 = 11, 0.... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
SUT
S
T
(4) Since the part of S not in Int S lies in Bd S,
it has content zero. Then additivity implies that
area S = area(1nt S) + area (S - Int S)
= area (Int S) .
A similar remark shows that
area ( S u Bd SJ = area (Int S ) + area(Bd S)
= area (Int S ) .
1
Remark.
L e t
S be a bounded s e t i n t h e p... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
t h e numbers a ( P ) , a s P
and d e f i n e t h e o u t e r -a r e a o f S t o b e t h e infemum of t h e numbers
A ( P ) . If t h e i n n e r a r e a and o u t e r a r e a of S a r e e q u a l , t h e i r
common v a l u e i s c a l l e d t h e -a r e a of S.
W e leave it a s a ( n o t t o o d i f f i c u l t ) e... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
r a l .
I
Eh ercises
1. Show t h a t i f ISS 1 e x i s t s , then Bd S hz-s content zero.
[Hint: Chwse Q s o t h a t S C Q . Since S& IS e x i s t s , t h e r e a r e functions
s and t t h a t a r e s t e p functions r e l a t i v e t o a p a r t i t i o n P of Q,
such
t h a t s <, Is jt o
Q and
[$ ( t - s ) <... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
r f a c e s z = xy and z = 0 and
Let Q denote the rectangle [0,1] x [0,1] in the following exercises.
@(a) Let f(x,y) = l/(y-x)
if x # y,
f(x,y) = 0
Does JJQf exist?
if x = y.
(b) Let g(x,y) = sin (l/(y-x))
if x # y,
Does JJQg exist?
@ ~ e t f(x,y) = 1if x = 112 and y is rational,
,-
f(x,y) = 0 otherwise
S... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/2b28a77b688c07a5c2d460b828ae1703_MIT18_024s11_ChDnotes.pdf |
T Y
R
E
B
I
L
4
199
Source: Wikipedia
Jacob (James) Bernoulli
(1654–1705)
For Bernoulli Trials
Figure by MIT OCW.
ESD.86
Class #2
February 12, 2007
Analyzing a Probability Problem
Four Steps to Happiness
1. Define the Random Variable(s)
2. Identify the (joint) sample space
3. Determine the probability law over the
... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/2b55649e3a413f0a06ad03e0cc6d7c16_lec2.pdf |
who pass by up to and
including the one who is the kth person interviewed.
P{Y=y}=P{exactly k-1 interviews occur in (y-1)
people passing AND the yth person passing is
Negative Binomial Probability Mass Function
interviewed}
Indicator random variables.
Suppose
Then E[Xi] = 1*pi + 0*(1- pi) = pi .
Example 1: Flip a c... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/2b55649e3a413f0a06ad03e0cc6d7c16_lec2.pdf |
= the number of hats returned to the correct owners. Then,
Answer independent of the number of players or teams!
Example 2. Baseball Hats.
Suppose that one player from each of the 30 Major League Baseball
teams attends a party at MIT, and each arrives wearing his
team’s baseball cap. Each tosses his hat into a closet... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/2b55649e3a413f0a06ad03e0cc6d7c16_lec2.pdf |
(1738; English trans. 1954)
‘Google’ this & find many interesting articles, such as
http://plato.stanford.edu/entries/paradox-stpetersburg/
Does the St. Petersburg Paradox Occur in Nature?
A possible term project!
Size and frequency of occurrence of Earthquakes.
Small earthquakes occur every day all around the world, ... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/2b55649e3a413f0a06ad03e0cc6d7c16_lec2.pdf |
grenade
Construction site blast
WWII conventional
bombs
late WWII conventional
bombs
WWII blockbuster bomb
Massive Ordnance Air
Blast bomb
Chernobyl nu clear
disaster, 1986
Small atomic bomb
Average tornado (total
energy)
Nagasaki atomic bomb
Little Skull Mtn., NV
Quake, 1992
Double Spring Fl at, NV
Quake, 1994
Northri... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/2b55649e3a413f0a06ad03e0cc6d7c16_lec2.pdf |
Let T12 = event successful Transmission from node 1 to node 2.
Assume all links function independently.
Then,
P{T12}=P{E or (AB) or (AC) or (DB) or (DC)}
P{T12}=P{E + E’[(AB) + (AC) + (DB) + (DC)]}
P{T12}=P{E + E’[A(B+C) + D(B +C)]}
P{T12}=P{E + E’[(A+D)(B+C)]}= P{E + E’[(A+A’D)(B+B’C)]}
P{T12}= pE + (1-pE){[pA+(1- ... | https://ocw.mit.edu/courses/esd-86-models-data-and-inference-for-socio-technical-systems-spring-2007/2b55649e3a413f0a06ad03e0cc6d7c16_lec2.pdf |
%jacobian2by2.m
%Code 8.1 of Random Eigenvalues by Alan Edelman
%Experiment: Compute the Jacobian of a 2x2 matrix function
%Comment: Symbolic tools are not perfect. The author
% exercised care in choosing the variables.
syms p q r s a b c d t e1 e2
X=[p q ; r s]; A=[a b;c d];
%% Compute Jacobians
Y=X^2; J=jacob... | https://ocw.mit.edu/courses/18-996-random-matrix-theory-and-its-applications-spring-2004/2b7145efcda0c3575006b63814358991_jacobian2by2.pdf |
Intro
Administrivia.
• Signup sheet.
• prerequisites: 6.046, 6.041/2, ability to do proofs
• homework weekly (first next week)
•
collaboration
• independent homeworks
• grading requirement
• term project
•
books.
• question: scribing?
Randomized algorithms: make random choices during run. Main benefits:
• speed... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/2b92be49ad6bbf6b75da5d5c262a27f1_n1.pdf |
versarial” may mean “well structured” i.e. natural
• fingerprinting/verification
– generate short random fingerprints for things
– faster than comparing things
– almost every fingerprint works
– so a random one works
2
• random sampling. graph algs, computational geometry, median
– fast way to find “typical” member... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/2b92be49ad6bbf6b75da5d5c262a27f1_n1.pdf |
i + 1) (may have outer elements)
• analysis:
n
n
pij ≤
�
�
i=1 j>i
2/(j − i + 1)
�
�
i=1 j>i
n n−i+1
2/k
1/k
=
�
�
i=1 k=1
n
n
≤ 2
�
�
i=1 k=1
≤ 2nHn
4
(Define Hn, claim O(log n).)
= O(n log n).
• analysis holds for every input, doesn’t assume random input
• we proved expected. can show... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/2b92be49ad6bbf6b75da5d5c262a27f1_n1.pdf |
)
≤
2Hn
�
u
• result: exists size O(n log n) auto
• gives randomized construction
• equally important, gives probabilistic existence proof of a small
BSP
• so might hope to find deterministically.
MinCut
• the problem
•
contraction
• conditionally independent events
• give/analyze
• repetition for better succ... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/2b92be49ad6bbf6b75da5d5c262a27f1_n1.pdf |
Why to Study
Finite Element Analysis!
That is, “Why to take 2.092/3”
Klaus-Jürgen Bathe
Why You Need to Study
Finite Element Analysis!
Klaus-Jürgen Bathe
Analysis is the key to
effective design
effective design
We perform analysis for:
• deformations and internal forces/stresses
• temperatures and heat tr... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/2ba586c8a074124bc6bd80ec6c93882e_MIT2_092F09_lec01.pdf |
of finite element computer
programs (such as NASTRAN,
ANSYS, ADINA, SIMULIA, etc…)
These analysis programs are
interfaced with computer-aided
,
) p g
design (CAD) programs Catia,
desi n CAD ro rams Catia
SolidWorks, Pro/Engineer, NX,
etc.
g (
The process of modeling for analysis
The process of modeling fo... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/2ba586c8a074124bc6bd80ec6c93882e_MIT2_092F09_lec01.pdf |
. model
Use of 3x3
16el. model
Use of 2x 2
16x64 element model
use of 3x3 Gauss
Gauss integration Gauss integration
integration
1
1
2
3
4
5
6
112.4
112.4
634.5
906.9
154 8
2654
2691
110.5
110.5
617.8
905.5
958.4 *
1528
2602
110.6
110.6
606.4
905.2
1441
2345
2664
*Spurious mode (phantom o... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/2ba586c8a074124bc6bd80ec6c93882e_MIT2_092F09_lec01.pdf |
7)
• Probably due to the Sleipner accident,
• Probabl due to the Slei ner accident
,
p
y
increased analysis attention was given to
critical components
– designers and analysts worked closely
together
Accuracy - part of reality
Coarse Mesh
Converged Mesh
Reference Mesh
Correct surface stress prediction at cr... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/2ba586c8a074124bc6bd80ec6c93882e_MIT2_092F09_lec01.pdf |
pled with structural
interactions –
an increasingly important analysis area
• Full Navier-Stokes equations for
incompressible or fully compressible flows
• Arbitrary Lagrangian-Eulerian formulation for
the fluid
Shock absorber
Shock absorber
Assembly parts
Shock absorber
Structural model
Shock absorber ... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/2ba586c8a074124bc6bd80ec6c93882e_MIT2_092F09_lec01.pdf |
Particle
trace plot
Analysis of an artificial lung
Particle trace
Radio-frequency tissue ablation
Electrode
Lesion
Courtesy of Medtronic, Inc. Used with permission.
Radio-frequency tissue ablation
Catheter
Electrode
Symmetry
face
Blood
Tissue
Radio-frequency tissue ablation
Temperature variation during... | https://ocw.mit.edu/courses/2-092-finite-element-analysis-of-solids-and-fluids-i-fall-2009/2ba586c8a074124bc6bd80ec6c93882e_MIT2_092F09_lec01.pdf |
ELECTRIC FORCES ON CHARGES
Lorentz Force Law:
+ × μ
)o
H Newtons
a = f/m = qE/m ≈ eV/mL [m s-2]
(
E v
=
q
f
Kinematics*:
t
v
=
a(t)dt
∫
0
=
v
o
+
ˆ
at
z
cathode
-V
-
E⊥
heated
filament
+
z
z = z + z•v t + at /2
o
ˆ
2
o
⇒
f
=
qE ma
=
anode, phosphors
deflection plates
cathode ray tube ... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/2c166c5e5dc71554eab3b62b6c9979aa_MIT6_013S09_lec05.pdf |
Attractive pressure
L5-2
ENERGY METHOD
FOR FINDING FORCES
Force, work, and energy:
dw = f ds ⇒ f =
dw
ds
[N]
C = εοA/s
1
w = CV2 =
2
2
2
1 Q s
1 Q
=
2 C 2 A
ε
o
[J]
=
f =
Q ≠ f(s) if C is open circuit
2
1 Q
dw
2 Aε
ds
o
2
( EA)
1(
ε
2
o
E )A
=
o
2 A 2
ε
o
= -PeA [N]
1 E
ε ... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/2c166c5e5dc71554eab3b62b6c9979aa_MIT6_013S09_lec05.pdf |
=
=
2
ε
2
A
= −
P A
e
*C = εA/s
L5-4
LATERAL FORCES – ENERGY METHOD
Energy derivative:
= −
f
(externally applied)
dw
dD
2
w
=
2
s
Q
Q
2C 2 WD
ε
o
=
W
E
+Q
C
A’ = Ws
Fringing
field
s
A’
f
D
-Q
=
f
2
Q s
=
2
2 WD
ε
o
2
( EWD)
ε
o
2 WD
ε
o
2
s
(
=
1
2
2
E
ε
o
)Ws
=
... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/2c166c5e5dc71554eab3b62b6c9979aa_MIT6_013S09_lec05.pdf |
2 R
ε
θ
o
=
2
E
ε
o
2
A'
[Nm]
≅
pressure
×
gap-area A'
×
θ
R
+
stator
rotor
-
A’ = 2Rs
R
2
-
+
T
Motor power:
Peak power:
Average power: Pavg = P/2 (duty cycle = ½)
P = Tω
[W]
n = 4
θ
Segmentation advantage:
T [Nm] = -dw/dθ ∝ A’ ∝ nRs (n = # segments)
v
+
-
rotor
stator
L5-6
ELEC... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/2c166c5e5dc71554eab3b62b6c9979aa_MIT6_013S09_lec05.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.917 Topics in Algebraic Topology: The Sullivan Conjecture
Fall 2007
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
The Adem Relations (Continued) (Lecture 5)
We continue to work with complexes over the finite field F2 with ... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2c1e5e606f38a085cc3c340d5290f675_lecture5.pdf |
let’s consider the case where V = F2 is a complex concentrated
in degree 0. In this case, we can identify VhΣ2 with the chain complex C (BΣ2), and we can identify V hΣ2
with the cochain complex C ∗(BΣ2). The norm map induces a map
∗
Hn(BΣ2)
→
H−n(BΣ2).
This is just the usual norm map in the theory of group cohomolo... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2c1e5e606f38a085cc3c340d5290f675_lecture5.pdf |
, t−1]/F2[t].
∗
Using this isomorphism, H (BΣ2) has a basis consisting of {tn}n<0. In previous lectures, we used a basis
{xi}i≥0 for H (BΣ2) which was dual to the basis {ti}i≥0 for H∗(BΣ2). By comparing degrees, we see that
these bases are related by the following transformation
∗
∗
xi �→ t−i−1
.
It follows that the ... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2c1e5e606f38a085cc3c340d5290f675_lecture5.pdf |
induced maps
→
V . We obtain
f : D2(F2)[−2n] � D2(F2[−n]) → D2(V )
f � : DT (F2)[−2n] � D2(F2[−n]) → DT (V ).
For every integer k, we let Sk(v) ∈ Hn+k(DT (V )) denote the image of tk−n ∈ Hk−n(DT (F2)) under the map
f �. If k ≥ n, then
tk−n ∈ Hk−n(D2(F2)) ⊆ Hk−n(DT (F2)).
In this case, we will denote the image o... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2c1e5e606f38a085cc3c340d5290f675_lecture5.pdf |
on V :
(∗) The cohomology groups Hn(V ) are finite dimensional for every n ∈ Z, and vanish for n sufficiently
small.
Assuming condition (∗), we have equivalences
V hΣ2 � V ⊗ (F2)hΣ2
V T Σ2 � V ⊗ (F2)T Σ2
VhΣ2 � V ⊗ (F2)hΣ2 .
Passing to cohomology, we obtain isomorphisms
H∗(V hΣ2 ) � H∗(V )[t]
H∗(V T Σ2 ) � H∗(V )[... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2c1e5e606f38a085cc3c340d5290f675_lecture5.pdf |
fies tmSk(v) = Sm+k(v).
→
⇒
2
The coefficient of tk−l in φ(Sk(v)) is given by
Res(tl−k−1φ(Sk(v))) = Res(φ(Sl−1(v))).
3
�
�
�
�
�
We have a commutative diagram
H∗(V ) Sl−1
� �
H∗(DT (V ))
� �
id
H∗(
V )
Sql �
� H∗(D
2(V ))
� H∗(V )[t, t−1
H∗(V )[t, t−1]
�������������
Res
]/ H∗(V )[t]
Res
� H∗(V ).
We now ... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2c1e5e606f38a085cc3c340d5290f675_lecture5.pdf |
the following:
Corollary 4. The inclusion j : Σ2 × Σ2 → G induces a restriction map on cohomology H∗(BG)
Σ2) � F2[t, u]. For k ≥ n, this map carries Sk(un) ∈ Hm+k(BG) to
→
H∗(Σ2 ×
�
(n − l, l)u n+ltk−l .
p
We observe that H (BG) � H−∗(D2(C (BΣ2))) has a basis consisting of products {xixj }0≤i<j and
xi}0≤i≤n. We o... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2c1e5e606f38a085cc3c340d5290f675_lecture5.pdf |
complete the calculation of the last lecture. Recall that we need to show that for
l
p, q > 0, the homology classes
�
(p − 2l, l) Sq
xp−l ∈ Hp+q(BG)
−q−l
l
4
�
�
�
�
�
�
�
�
�
�
�
(q − 2l�, l�) Sq
−p−l�
xq−l� ∈ Hp+q(BG)
l�
have the same image in H (BΣ4). Invoking Corollary 5, we see that it suffices to show ... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/2c1e5e606f38a085cc3c340d5290f675_lecture5.pdf |
6.890 Algorithmic Lower Bounds and Hardness Proofs
Fall 2014
Erik Demaine
Class 2 Scribe Notes
1 Useful Problems for Hardness Reductions
This lecture mostly focuses on using 3-Partition to solve 2+ problems by reducing to number problems.
The basic idea is to think of your numbers as integers – fixed-point or rati... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
S ⊂ A such that ΣS = t. It is easy to think of
an instance of this problem as a partition, although it’s a generalization. Reducing from Subset Sum that
we can reduce from 2-Partition. 2-Partition to Subset Sum is a strict generalization – not given t –
but we are essentially choosing a subset A1 whose sum is ΣA/2. ... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
A), but all the ai’s become roughly equal to each other and arbitrarily close to t/3. Let’s talk
like n
about a couple of related problems to 3-Partition, and why we’re talking about it.
100
1.2.1 Numerical 3-Dimensional Matching
This problem has a funny name – we will get to that second, don’t worry for the moment.... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
× max(A ∪ B ∪ C). The new tf becomes t + 13∞. This forces us to pick one from A, one
from B, and one from C. We must show that the infinities can be treated algebraically despite being so
large. Note that we require that the sets be of size exactly 3 as part of the 3-Partition specification for
this reduction.
1.2.2 ... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
to Num 3-Dim Matching). Because tripartite hypergraphs are
3-uniform, X3C is a strict generalization of 3DM.
1.3 Strong vs. Weak
Back to the issue of 2-Partition vs. 3-Partition, we explore an important distinction – weak vs. strong
NP-hardness. There are two types of NP-hardness for number problems when we have in... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
But today, we’re going to use unary a lot to talk about Strongly NP-hard
problems. So, note: Strong NP-hardness is “My problem is so hard that even if I encode my numbers in
unary, it’s still NP-hard.” If you can prove Strong, you should. It’s better. Of course Weak NP-hardness is
still okay.
1.4 Pseudopolynomial, ... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
general, strongly NP-hard is a better result, because it is more restrictive.
• 2-Partition is weakly NP-hard and Pseudopolynomial (therefore not strongly NP-hard)
• 3-Partition is strongly NP-hard (therefore not even Pseudopolynomial)
3
2 Reductions!
Let’s do some reductions, shall we? It’s NP-hardcore time! Jus... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
Let’s look at a special case: packings should be exact –
no gaps. Rotation must be integer ×90◦, so constant number of them. Translations are integral also given
exact packing , so there are succinct encodings implying that this problem is ∈NP. We now prove that both
of these problems are strongly NP-hard.
2.2.1 Re... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
n unit squares (tiles), each
one with 4 colors (a, b, c, d), with one color per edge; and also a target rectangle, and we want to pack the
squares into the rectangle with colors matching. Eternity II, is also an edge matching puzzle, currently still
unsolved, which for a while had US $2,000,000 in prize money; this ... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
n/3
t
$
$
$
%
%
%
%
%
5
Are there any numbers in this problem, as inputs? The colors need to be represented as numbers
(although they are only compared, never added). Total number of different colors ≤ 4n, anyways. So it
wouldn’t make sense to say this is Strongly NP-hard because the problem can really onl... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
colors are unique pairs to force their assembly. Through this reduction, we
see that the hardness is the same for Signed and Unsigned Edge-Matching puzzles.
2.5 Jigsaw Puzzles
For Jigsaw Puzzles, each edge is either straight, pocket, or tab. Pockets and tabs can be slightly different
in shape. We can allow for some ... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
use log 4n bits along the edge by indent or outdenting accordingly (See Slide
13). They will fit together and make a nice clean seam if the colors match. The construction is blown up by
a logarithmic factor log 4n, but the number of pieces stays the same. Now, we reduce Polyomino Packing to
Edge-Matching ... by makin... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
of into a
rectangle, we create a rectangle gadget (See Slide 14). Make it so that the extra space is a rectangle and
then scale up by 3B + t. If you want to completely pack the target square, you can compute how much extra
slop there is and add in a bunch of 1x1 tiles so that you can fill in the extra space, making i... | https://ocw.mit.edu/courses/6-890-algorithmic-lower-bounds-fun-with-hardness-proofs-fall-2014/2c3913aefe7be0f98fcc16394ef6ea4b_MIT6_890F14_Lec2.pdf |
Chapter 8
Meson Mass Matrix
Close up on meson mass matrix states as:
Vm = υ
+
1 n
f √6
)2
mu +
+
1 π◦
8 √2
1 π◦ +
1 a
F √3
(
{
1 a
F √3 − 8 √2
1 a
1 2n
F √3 − f √6
υ
2f 2
a π◦ η
(
(
1 n
f √6
)2 ms}
�
�
f 2 2
F 2 3(mu + md + ms)
−
√6
2f mu+md−
F
3√2
2f mu
F
2ms
md
⎛
⎜
⎜
⎝
⎛
⎜
⎝
a
π◦
η
⎞
⎟
⎠... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/2c3cf052ae853f5b31fa7246895672d6_chap8.pdf |
+md−
F
3√2
1
3(mu −
√
1
+ md + 4ms)
3 (mu
md)
⎞
⎟
⎟
⎠
1 (axions). There is a small eigenvalue
associated with an
f 2
F 2
∼
eigenvalue
∼
⎛
⎜
⎝
1
f
aF
f bF ⎟
⎞
⎠
with
60
=
=
⎛
⎜
⎜
⎝
�
So
∈
M
υ
2 )
( f
1. For f
F
61
+ (mu + md)a +
1
√3
(mu −
md)b = 0(8.4)
md
2
mu −
√6
1
√3
2ms) +
√2
3
(mu... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/2c3cf052ae853f5b31fa7246895672d6_chap8.pdf |
we take mu = md = 0
∈
M
υ
( f
2 )
=
⎛
⎜
⎜
⎝
f 2
F 2
2
3 ms
0
0
0
4
3√2 ms 0
f
F −
f
4
3√2 ms
F −
0
4
3ms
⎞
⎟
⎟
⎠
(8.6)
(8.7)
(8.8)
(8.9)
(8.10)
has two vanishing eigenvalues. So η gets infected and the GM-O relation is
badly violated. The general case is a little messy but with mu = md �
ms w... | https://ocw.mit.edu/courses/8-325-relativistic-quantum-field-theory-iii-spring-2003/2c3cf052ae853f5b31fa7246895672d6_chap8.pdf |
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.341: Discrete-Time Signal Processing
OpenCourseWare 2006
Lecture 7
IIR, FIR Filter Structures
Reading: Sections 6.1 - 6.5 in Oppenheim, Schafer & Buck (OSB).
Signal Flow Graphs
A linear time-invariant discrete-time ... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2c6fe95a4d44834ccf0c0d9017118703_lec07.pdf |
signal glow graph corresponding to the first order system in
OSB Figure 6.10. By convention, the delay element has been represented by a branch gain of
z−1 .
The signal flow graph representation of a LTI system is not unique. In fact, for any given
rational system function, equivalent sets of difference equations and ... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2c6fe95a4d44834ccf0c0d9017118703_lec07.pdf |
to physical memories in actual implementation, direct form
II structures require less state memory than the direct form I implementation. However, the
total memory requirement for both forms are similar, because direct form II structures need
more cache memory during computations.
Transposed Forms
Using signal flow... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2c6fe95a4d44834ccf0c0d9017118703_lec07.pdf |
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